Note that the use of the Sinkhorn divergence (Definition REF ) is also interesting from a statistical perspective.
It is proved in [1]} that for balanced OT, estimating \((,)\)  (not \(\) ) using \((_N,_N)\)  when \(x,y)={x-y}_2^2\)  and \(\ne \)  yields an estimation error scaling as \(O(N^{-2/d})\) . This matches the rates of the (unregularized) estimator, but leads to a faster computational scheme thanks to Sinkhorn's iterations.
